Profinite Groups Associated to Sofic Shifts are Free

We show that the maximal subgroup of the free profinite semigroup associated by Almeida to an irreducible sofic shift is a free profinite group, generalizing an earlier result of the second author for the case of the full shift (whose corresponding maximal subgroup is the maximal subgroup of the minimal ideal). A corresponding result is proved for certain relatively free profinite semigroups. We also establish some other analogies between the kernel of the free profinite semigroup and the \J-class associated to an irreducible sofic shift

Equidivisible pseudovarieties of semigroups

We give a complete characterization of pseudovarieties of semigroups whose finitely generated relatively free profinite semigroups are equidivisible. Besides the pseudovarieties of completely simple semigroups, they are precisely the pseudovarieties that are closed under Mal'cev product on the left by the pseudovariety of locally trivial semigroups. A further characterization which turns out to be instrumental is as the non-completely simple pseudovarieties that are closed under two-sided Karnofsky-Rhodes expansion

On the group of a rational maximal bifix code

We give necessary and sufficient conditions for the group of a rational maximal bifix code $Z$ to be isomorphic with the $F$-group of $Z\cap F$, when $F$ is recurrent and $Z\cap F$ is rational. The case where $F$ is uniformly recurrent, which is known to imply the finiteness of $Z\cap F$, receives special attention. The proofs are done by exploring the connections with the structure of the free profinite monoid over the alphabet of $F$

A profinite approach to complete bifix decodings of recurrent languages

We approach the study of complete bifix decodings of (uniformly) recurrent languages with the help of the free profinite monoid. We show that the complete bifix decoding of a uniformly recurrent language $F$ by an $F$-charged rational complete bifix code is uniformly recurrent. An analogous result is obtained for recurrent languages.Comment: Original Manuscript of article to be published by De Gruyter in Forum Mathematicum. The last section of the version in Forum Mathematicum is very different, as there it is not proved that the Sch\"utzenberger group is an invariant of eventual conjugacy (the argument in the Original Manuscript had a flaw), but only that its maximal pronilpotent quotient is invariant by eventual conjugac

A new algebraic invariant for weak equivalence of sofic subshifts

It is studied how taking the inverse image by a sliding block code affects the syntactic semigroup of a sofic subshift. Two independent approaches are used: ζ-semigroups as recognition structures for sofic subshifts, and relatively free profinite semigroups. A new algebraic invariant is obtained for weak equivalence of sofic subshifts, by determining which classes of sofic subshifts naturally defined by pseudovarieties of finite semigroups are closed under weak equivalence. Among such classes are the classes of almost finite type subshifts and aperiodic subshifts. The algebraic invariant is compared with other robust conjugacy invariants.Research programme AutoMathA of ESF; Pessoa bilateral project Egide/Grices 11113YM "Automata, profinite semigroups and symbolic dynamics"; FCT, grant SFRH/BD/24200/2005; POCI 2010; FS

A new algebraic invariant for weak equivalence of sofic subshifts

It is studied how taking the inverse image by a sliding block code affects the syntactic semigroup of a sofic subshift. Two independent approaches are used: ζ-semigroups as recognition structures for sofic subshifts, and relatively free profinite semigroups. A new algebraic invariant is obtained for weak equivalence of sofic subshifts, by determining which classes of sofic subshifts naturally defined by pseudovarieties of finite semigroups are closed under weak equivalence. Among such classes are the classes of almost finite type subshifts and aperiodic subshifts. The algebraic invariant is compared with other robust conjugacy invariants.Research programme AutoMathA of ESF; Pessoa bilateral project Egide/Grices 11113YM "Automata, profinite semigroups and symbolic dynamics"; FCT, grant SFRH/BD/24200/2005; POCI 2010; FS